Exponential Dichotomy for a Nonautonomous System of Parabolic Equations

In this paper we study the existence and roughness of exponential dichotomy (ED) of a non-autonomous system of parabolic equations with Neumann boundary conditions. In order to do that, we first set the problem in the Linear Skew-Product Semiflow (LSPS) framework. Then we prove that the ED is not destroyed by small perturbation (roughness). Next, we compute the dynamical spectrum for this LSPS. Finally, under some conditions we prove that zero does not belong to the dynamical spectrum corresponding to this LSPS. i.e., the system has ED (existence).